The research group Geometry, Algebra and Physics in Deep Neural Networks (GAPinDNNs) at the Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, develops mathematical foundations for deep learning. Our research combines theory with large-scale numerical experiments: theoretical predictions about neural networks must be tested empirically, and new architectures derived from mathematical principles must be validated on realistic benchmarks. This project requests GPU resources on Arrhenius for the group’s computational research, organized in the following lines of work. Scaling laws and finite-width theory. We have developed a Feynman diagram framework for computing finite-width corrections to the statistics of neural networks and their training dynamics. In the current project, we study how neural networks scale with width, depth and dataset size, and compare the empirical scaling behavior with predictions from our theory. This requires training large families of networks across many model sizes and random seeds. Equivariant neural networks. Neural networks that respect the symmetries of their data are central to applications in science. We will systematically compare and benchmark equivariant architectures based on regular representations with those based on irreducible representations, and study how their performance and computational cost scale with model size. We will also continue our work on achieving equivariance through data augmentation, including stochastic weight averaging and Bayesian neural networks, which requires extensive experiments on vision and graph classification models. Geometric learning on point clouds and molecules. We have recently introduced neural point-forms, a method based on diffusion geometry that learns differential forms from point clouds. We will evaluate this method on further benchmarks, starting with spatial transcriptomics classification. The main computational effort will go into extending the method to molecular energy prediction on OMol25, one of the largest available datasets of quantum chemical calculations, with more than 100 million DFT calculations covering a wide range of molecular systems. Molecular energies are invariant under rotations, so this requires developing new ways of incorporating rotational symmetry into the diffusion geometry framework. Training and comparing these variants at the scale of OMol25 is computationally demanding. All projects use PyTorch-based training pipelines, run mostly as single-GPU jobs organized in large arrays for parameter sweeps, model-size scans and repeated runs with different seeds, with occasional multi-GPU runs for the largest models. We request 5000 GPU-hours per month on Arrhenius GPU, together with 10,000 GiB of project storage and 1 million files for datasets, model checkpoints and experiment outputs. The results will be published at leading machine learning conferences and journals and will contribute to the PhD and MSc theses in the group.